Lemma 1

If $\sigma \in S_n$ is the permutation such that $\sigma(h) = i_h$ for $1 \leq h \leq p$, $\sigma(h) = j_{n-p}$ for $p + 1 \leq h \leq n$, then $\varepsilon_\sigma = (-1)^\nu$.

For $1 \leq h < h' \leq p$ or $p + 1 \leq h < h' \leq n$, $\sigma(h') > \sigma(h)$ and the number of ordered pairs $(h, h')$ such that $1 \leq h \leq p < h' \leq n$ and $\sigma(h) > \sigma(h')$ is equal to $\nu$.